Function which performs the fitting of an adaptive mixture of
Student-t distributions to approximate a target density through its
kernel function

Usage

1

Arguments

KERNEL

kernel function of the target density on which the adaptive mixture is fitted. This
function should be vectorized for speed purposes (i.e., its first
argument should be a matrix and its output a vector). Moreover, the function must contain
the logical argument log. If log = TRUE, the function
returns (natural) logarithm values of the kernel function. NA and
NaN values are not allowed. (See *Details* for examples
of KERNEL implementation.)

mu0

initial value in the first stage optimization (for the location of
the first Student-t component) in the adaptive mixture, or
location of the first Student-t component if Sigma0 is not NULL.

Sigma0

scale matrix of the first Student-t component (square, symmetric and positive definite). Default:
Sigma0 = NULL, i.e., the scale matrix of the first Student-t
component is estimated by the function AdMit.

control

control parameters (see *Details*).

...

further arguments to be passed to KERNEL.

Details

The argument KERNEL is the kernel function of the target
density, and it should be vectorized for speed purposes.

As a first example, consider the kernel function proposed by Gelman-Meng (1991):

This way, we may supply a point (x1,x2)
for x and the function will output a single value (i.e., the kernel
estimated at this point). But the function is vectorized, in the sense
that we may supply a Nx2 matrix
of values for x, where rows of x are
points (x1,x2) and the output will be a vector of
length N, containing the kernel values for these points.
Since the AdMit procedure evaluates KERNEL for a
large number of points, a vectorized implementation is important. Note
also the additional argument log = TRUE which is used for
numerical stability.

As a second example, consider the following (simple) econometric model:

y_t ~ i.i.d. N(mu,sigma^2) t=1,...,T

where mu is the mean value and sigma is the
standard deviation. Our purpose is to estimate
theta=(mu,sigma) within a Bayesian
framework, based on a vector y of T observations; the
kernel thus consists of the product of the
prior and the likelihood function. As previously mentioned, the
kernel function should be vectorized, i.e., treat a (Nx2) matrix of points
theta for which the kernel will be evaluated.
Using the common (Jeffreys) prior p(theta)=1/sigma
for sigma>0, a vectorized implementation of the
kernel function might be:

KERNEL <-function(theta, y,log=TRUE){if(is.vector(theta))
theta <-matrix(theta,nrow=1)## sub function which returns the log-kernel for a given## thetai value (i.e., a given row of theta)
KERNEL_sub <-function(thetai){if(thetai[2]>0)## check if sigma>0{## if yes, compute the log-kernel at thetai
r <--log(thetai[2])+sum(dnorm(y, thetai[1], thetai[2],TRUE))}else{## if no, returns -Infinity
r <--Inf}as.numeric(r)}## 'apply' on the rows of theta (faster than a for loop)
r <-apply(theta,1, KERNEL_sub)if(!log)
r <-exp(r)as.numeric(r)}

Since this kernel function also depends on the vector y, it
must be passed to KERNEL in the AdMit function. This is
achieved via the argument ..., i.e., AdMit(KERNEL, mu = c(0, 1), y = y).

To gain even more speed, implementation of KERNEL might rely on C or Fortran
code using the functions .C and .Fortran. An example is
provided in the file ‘AdMitJSS.R’ in the package's folder.

The argument control is a list that can supply any of
the following components:

Ns

number of draws used in the evaluation of the
importance sampling weights (integer number not smaller than 100). Default: Ns = 1e5.

Np

number of draws used in the optimization of the mixing
probabilities (integer number not smaller than 100 and not larger
than Ns). Default: Np = 1e3.

Hmax

maximum number of Student-t components in the
adaptive mixture (integer number not smaller than one). Default: Hmax = 10.

df

degrees of freedom parameter of the
Student-t components (real number not smaller than one). Default: df = 1.

CVtol

tolerance for the relative change of the coefficient of
variation (real number in [0,1]). The
adaptive algorithm stops if the new
component leads to a relative change in the coefficient of
variation that is smaller or equal than
CVtol. Default: CVtol = 0.1, i.e., 10%.

weightNC

weight assigned to the new
Student-t component of the adaptive mixture as
a starting value in the optimization of the mixing probabilities
(real number in [0,1]). Default: weightNC = 0.1, i.e., 10%.

trace

IS

should importance sampling be used to estimate the
mode and the scale matrix of the Student-t components (logical). Default: IS = FALSE,
i.e., use numerical optimization instead.

ISpercent

vector of percentage(s) of largest weights used to
estimate the mode and the scale matrix of the Student-t
components of the adaptive mixture by importance
sampling (real number(s) in [0,1]). Default:
ISpercent = c(0.05, 0.15, 0.3), i.e., 5%, 15% and 30%.

ISscale

trace.mu

Tracing information on
the progress in the optimization of the mode of the mixture
components (non-negative integer number). Higher values
may produce more tracing information (see the source code
of the function optim for further details).
Default: trace.mu = 0, i.e., no tracing information.

maxit.mu

maximum number of iterations used
in the optimization of the modes of the mixture components
(positive integer). Default: maxit.mu = 500.

reltol.mu

relative convergence tolerance
used in the optimization of the modes of the mixture components
(real number in [0,1]). Default: reltol.mu = 1e-8.

trace.p, maxit.p, reltol.p

the same as for
the arguments above, but for the optimization of the mixing
probabilities of the mixture components.

where H (>=1) is the number of components in the adaptive
mixture of Student-t distributions and d (>=1) is
the dimension of the first argument in KERNEL.

summary: data frame containing information on the optimization
procedures. It returns for each component of the adaptive mixture of
Student-t distribution: 1. the method used to estimate the mode
and the scale matrix of the Student-t component (‘USER’ if Sigma0 is
provided by the user; numerical optimization: ‘BFGS’, ‘Nelder-Mead’;
importance sampling: ‘IS’, with percentage(s) of importance weights
used and scaling factor(s)); 2. the time required for this optimization;
3. the method used to estimate the mixing probabilities
(‘NLMINB’, ‘BFGS’, ‘Nelder-Mead’, ‘NONE’); 4. the time required for this
optimization; 5. the coefficient of variation of the importance
sampling weights.

Note

Further details and examples of the R package AdMit
can be found in Ardia, Hoogerheide, van Dijk (2009a,b). See also
the package vignette by typing vignette("AdMit").

Further details on the core algorithm are given in
Hoogerheide (2006), Hoogerheide, Kaashoek, van Dijk (2007) and
Hoogerheide, van Dijk (2008).

The adaptive mixture mit returned by the function AdMit is used by the
function AdMitIS to perform importance sampling using
mit as the importance density or by the function AdMitMH to perform
independence chain Metropolis-Hastings sampling using mit as the
candidate density.

Please cite the package in publications. Use citation("AdMit").

Author(s)

David Ardia for the R port,
Lennart F. Hoogerheide and Herman K. van Dijk for the AdMit algorithm.

See Also

AdMitIS for importance sampling using an
adaptive mixture of Student-t distributions as the importance density,
AdMitMH for the independence chain Metropolis-Hastings
algorithm using an adaptive mixture of Student-t distributions as
the candidate density.